# Hint #27: Number of Squares on a Chessboard

27. How many squares are there on a chessboard or checkerboard?? (the answer is not 64)

Can you extend your technique to calculate the number of rectangles on a chessboard?

Lets first establish what we mean by a square to be clear why the answer is not 64. Below are some examples of valid squares.

The method I used was to think about a square of a specific size say a 2x2. And consider the number of locations that square can go. There is a simple relationship between the size of a square and the number of locations it can be placed.

Moving on to rectangles the same technique can still be used. Although it's a lot more fiddly. A better solution involves counting the number of available diagonals.

Where next?

© Nigel Coldwell 2004 -  – The questions on this site may be reproduced without further permission, I do not claim copyright over them. The answers are mine and may not be reproduced without my expressed prior consent. Please inquire using the link at the top of the page. Secure version of this page.

PayPal
I always think it's arrogant to add a donate button, but it has been requested. If I help you get a job though, you could buy me a pint! - nigel